Property Testing: A Learning Theory Perspective

Abstract

Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be performed by observing only a very small part of the object, in particular by querying the object, and the algorithm is allowed a small failure probability.One view of property testing is as a relaxation of learning the object (obtaining an approximate representation of the object). Thus property testing algorithms can serve as a preliminary step to learning. That is, they can be applied in order to select, very efficiently, what hypothesis class to use for learning. This survey takes the learning-theory point of view and focuses on results for testing properties of functions that are of interest to the learning theory community. In particular we cover results for testing algebraic properties of functions such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more.

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“…use, what range of ε to allow (the larger ε, the easier it should be to test P), and how to measure the complexity of the testing algorithm. A lot of work in classical computer science has gone into the study of efficient testers for various properties, as well as proofs that certain properties are not efficiently testable, see for instance [39,76,66,147,75]. Typically, X will be the set of all strings of length N over some finite alphabet, where we think of N as being very large.…”

Property Testing Lower Bounds via Communication Complexity

“…The notion of testing Boolean functions in this framework goes back to the seminal work of Rubinfeld & Sudan (1996) and has several connections to complexity theory (in particular PCPs and hardness of approximation), as well as computational learning theory (Ron 2008). Over the last two decades, researchers have exerted a considerable amount of effort to determine the query complexity for testing properties of a function f , such as whether f is a linear function (Blum et al 1993), whether f is isomorphic to a given function (Alon & Blais 2010;Blais & O'Donnell 2010;Chakraborty et al 2011b), whether f is a k-junta (Blais 2008(Blais , 2009Fischer et al 2004), a monotone function (Fischer et al 2002;Goldreich et al 2000), a dictator (Parnas et al 2002), a halfspace (Matulef et al 2009), an s-sparse polynomial, a size-s decision tree.…”

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“…Much research has been done on understanding the relation between property testing algorithms and learning algorithms, see, e. g., [16,18] and the lengthy survey [24]. As Goldreich has noted [15], an often-invoked motivation for property testing is that (inexpensive) testing algorithms can be used as a "preliminary diagnostic" to determine whether it is appropriate to run a (more expensive) learning algorithm.…”